{"id":46,"date":"2023-11-08T13:42:10","date_gmt":"2023-11-08T13:42:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/introduction-inverse-functions\/"},"modified":"2024-08-01T21:13:05","modified_gmt":"2024-08-01T21:13:05","slug":"7-2-inverse-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/7-2-inverse-functions\/","title":{"raw":"7.2 Inverse Functions","rendered":"7.2 Inverse Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3 style=\"text-align: center;\">Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Given a table or graph, determine if a function is one-to-one.<\/li>\r\n \t<li>Find the inverse of a linear function.<\/li>\r\n \t<li>Determine if two functions are inverses by finding [latex]f(f^{-1}(x))[\/latex] and [latex]f^{-1}(f(x))[\/latex].<\/li>\r\n \t<li>Graph a linear function and its inverse on the same axes.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"mceTemp\"><\/div>\r\n<h2>One-to-One Functions<\/h2>\r\nRecall that in a function, the input value must have one and only one value for the output.\u00a0The set of input values is the <strong>domain of the function<\/strong> and the set of output values is the <strong>range of the function<\/strong>.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the domain and range for the function.\r\n<table style=\"height: 134px; width: 348px;\" width=\"409\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center; width: 257.683px;\"><strong><i>[latex]x[\/latex]<\/i><\/strong><\/th>\r\n<th style=\"text-align: center; width: 224.95px;\"><strong><i>[latex]y[\/latex]<\/i><\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center; width: 257.683px;\">[latex]\u22125[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 224.95px;\">[latex]\u22126[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 257.683px;\">[latex]\u22122[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 224.95px;\">[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 257.683px;\">[latex]\u22121[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 224.95px;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 257.683px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 224.95px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 257.683px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 224.95px;\">[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"130987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"130987\"]\r\n\r\nThe domain is the set of inputs or [latex]x[\/latex]-coordinates.\r\n<p align=\"center\">[latex]\\{\u22125,\u22122,\u22121,0,5\\}[\/latex]<\/p>\r\nThe range is the set of outputs or\u00a0[latex]y[\/latex]-coordinates.\r\n<p align=\"center\">[latex]\\{\u22126,\u22121,0,3,15\\}[\/latex]<\/p>\r\n<p align=\"center\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nSome functions have a given output value that corresponds to two or more input values. For example, in the following stock chart the stock price was\u00a0[latex]$1000[\/latex] on five different dates, meaning that there were five different input values that all resulted in the same output value of\u00a0[latex]$1000[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200506\/CNX_Precalc_Figure_01_00_001n2.jpg\" alt=\"Figure of a bull and a graph of market prices.\" width=\"975\" height=\"307\" \/>\r\n<p id=\"fs-id1165135678633\">However, some functions have only one input value for each output value as well as having only one output value for each input value. We call these functions <strong>one-to-one functions<\/strong>. As an example, consider a school that uses only letter grades and decimal equivalents as listed below.<\/p>\r\n\r\n<table style=\"width: 353px;\" summary=\"Two columns and five rows. The first column is labeled,\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center; width: 234.55px;\">Letter grade<\/th>\r\n<th style=\"text-align: center; width: 278.717px;\">Grade point average<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center; width: 234.55px;\">A<\/td>\r\n<td style=\"text-align: center; width: 278.717px;\">[latex]4.0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 234.55px;\">B<\/td>\r\n<td style=\"text-align: center; width: 278.717px;\">[latex]3.0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 234.55px;\">C<\/td>\r\n<td style=\"text-align: center; width: 278.717px;\">[latex]2.0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 234.55px;\">D<\/td>\r\n<td style=\"text-align: center; width: 278.717px;\">[latex]1.0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137561844\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\r\nTo visualize this concept, look at the two simple functions sketched in (a) and (b) below. Note that (c) is not a function since the input\u00a0<em>q<\/em> produces two outputs,\u00a0<em>y<\/em> and\u00a0<em>z<\/em>.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200453\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/>\r\n\r\nThe function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>One-to-One Function<\/h3>\r\nA one-to-one function is a function in which each output value corresponds to exactly one input value.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine whether each of the following tables represents a one-to-one function.\r\n\r\na)\r\n<table style=\"width: 234px;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center; width: 235.45px;\"><strong>Input<\/strong><\/td>\r\n<td style=\"text-align: center; width: 247.183px;\"><strong>Output<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 235.45px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 247.183px;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 235.45px;\">[latex]12[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 247.183px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 235.45px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 247.183px;\">[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 235.45px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 247.183px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 235.45px;\">[latex]-5[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 247.183px;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nb)\r\n<table style=\"width: 234px;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center; width: 237.367px;\"><strong>Input<\/strong><\/td>\r\n<td style=\"text-align: center; width: 245.267px;\"><strong>Output<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 237.367px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 245.267px;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 237.367px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 245.267px;\">[latex]16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 237.367px;\">[latex]16[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 245.267px;\">[latex]32[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 237.367px;\">[latex]32[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 245.267px;\">[latex]64[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 237.367px;\">[latex]64[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 245.267px;\">[latex]128[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"945171\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"945171\"]\r\n\r\nTable a) maps the output value\u00a0[latex]2[\/latex] to two different input values, therefore\u00a0this is NOT a one-to-one function.\r\n\r\nTable b) maps each output to one unique input, therefore this IS a one-to-one function.\r\n\r\nOnly table b) is one-to-one.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show an example of using tables of values to determine whether a function is one-to-one.\r\n\r\nhttps:\/\/youtu.be\/QFOJmevha_Y\r\n<h2>Using the Horizontal Line Test<\/h2>\r\n<p id=\"fs-id1165137871503\">An easy way to determine whether a function\u00a0is a one-to-one function is to use the <strong>horizontal line test <\/strong>on the graph of the function. \u00a0To do this, draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nFor\u00a0the following graphs, determine which represent one-to-one functions.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200511\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/>\r\n[reveal-answer q=\"783411\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"783411\"]\r\n<p id=\"fs-id1165135185190\">The function in (a) is\u00a0not one-to-one. Using the horizontal line test, as shown below, it intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200515\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"\" width=\"294\" height=\"267\" \/>\r\n<figure id=\"Figure_01_01_010\" class=\"small\"><\/figure>\r\nThe function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.\r\n\r\n<img class=\"size-medium wp-image-2697 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212200\/Screen-Shot-2016-07-16-at-2.21.48-PM-287x300.png\" alt=\"Graph of a line with three dashed horizontal lines passing through it.\" width=\"287\" height=\"300\" \/>\r\n\r\n&nbsp;\r\n\r\nThe function (c) is not one-to-one and is in fact not a function.\r\n\r\n<img class=\" wp-image-2698 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212527\/Screen-Shot-2016-07-16-at-2.25.36-PM-237x300.png\" alt=\"Graph of a circle with two dashed lines passing through horizontally\" width=\"279\" height=\"353\" \/>\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video provides another example of using the horizontal line test to determine whether a graph represents a one-to-one function.\r\n\r\n[embed]https:\/\/youtu.be\/tbSGdcSN8RE[\/embed]\r\n<h2>Inverse Functions<\/h2>\r\nGiven a one-to-one function [latex]f\\left(x\\right)[\/latex], we represent its inverse as [latex]{f}^{-1}\\left(x\\right)[\/latex], read as \"[latex]f[\/latex] inverse of [latex]x[\/latex].\" The superscript [latex]-1[\/latex] is part of the notation. It is not an exponent; it does not imply a power of [latex]-1[\/latex] . In other words, [latex]{f}^{-1}\\left(x\\right)[\/latex] does <em>not<\/em> mean [latex]\\dfrac{1}{f\\left(x\\right)}[\/latex]. Instead, [latex]\\dfrac{1}{f\\left(x\\right)}[\/latex] is the reciprocal of [latex]f[\/latex], not the inverse.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Inverse Function<\/h3>\r\nFor any <strong>one-to-one function<\/strong> [latex]f\\left(x\\right)=y[\/latex], a function [latex]{f}^{-1}\\left(x\\right)[\/latex] is an <strong>inverse function<\/strong> of [latex]f[\/latex] if [latex]{f}^{-1}\\left(y\\right)=x[\/latex]. This can also be written as [latex]{f}^{-1}\\left(f\\left(x\\right)\\right)=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]. It also follows that [latex]f\\left({f}^{-1}\\left(x\\right)\\right)=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]{f}^{-1}[\/latex] if [latex]{f}^{-1}[\/latex] is the inverse of [latex]f[\/latex].\r\n\r\nThe notation [latex]{f}^{-1}[\/latex] is read \"[latex]f[\/latex] inverse.\" Like any other function, we can use any variable name as the input for [latex]{f}^{-1}[\/latex], so we will often write [latex]{f}^{-1}\\left(x\\right)[\/latex], which we read as [latex]\"f[\/latex] inverse of [latex]x[\/latex]\".\r\n\r\n<\/div>\r\n<h4 style=\"padding-right: 120px;\">Why -1?<\/h4>\r\n<p style=\"padding-right: 120px;\">The \"exponent-like\" notation comes from an analogy between function composition and multiplication: just as [latex]{a}^{-1}a=1[\/latex] for any nonzero number [latex]a[\/latex], so [latex]\\left({f}^{-1}\\circ f\\right)\\left(x\\right)={f}^{-1}\\left(f\\left(x\\right)\\right)={f}^{-1}\\left(y\\right)=x[\/latex]. This is true for all [latex]x[\/latex] in the domain of [latex]f[\/latex]. Informally, this means that inverse functions \"undo\" each other. However, just as zero does not have a <strong>reciprocal<\/strong>, some functions do not have inverses.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying an Inverse Function for a Given Input-Output Pair<\/h3>\r\nIf for a particular one-to-one function [latex]f\\left(2\\right)=4[\/latex] and [latex]f\\left(5\\right)=12[\/latex], what are the corresponding input and output values for the inverse function?\r\n\r\n[reveal-answer q=\"348517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"348517\"]\r\n\r\nThe inverse function reverses the input and output quantities, so if\r\n<p style=\"text-align: left;\">[latex]f\\left(2\\right)=4[\/latex], then [latex]{f}^{-1}\\left(4\\right)=2[\/latex]<\/p>\r\n<p style=\"text-align: left;\">and if<\/p>\r\n<p style=\"text-align: left;\">[latex]f\\left(5\\right)=12[\/latex], then [latex]{f}^{-1}\\left(12\\right)=5[\/latex]<\/p>\r\nAlternatively, if we want to name the inverse function [latex]g[\/latex], then [latex]g\\left(4\\right)=2[\/latex] and [latex]g\\left(12\\right)=5[\/latex].\r\n\r\nIf we show the coordinate pairs in a table form, it is clear that the the inputs and outputs are reversed.\r\n<table style=\"width: 315px;\" summary=\"For (x,f(x)) we have the values (2, 4) and (5, 12); for (x, g(x)), we have the values (4, 2) and (12, 5).\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center; width: 248.883px;\">[latex]\\left(x,f\\left(x\\right)\\right)[\/latex]<\/th>\r\n<th style=\"text-align: center; width: 233.75px;\">[latex]\\left(x,g\\left(x\\right)\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center; width: 248.883px;\">[latex]\\left(2,4\\right)[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 233.75px;\">[latex]\\left(4,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center; width: 248.883px;\">[latex]\\left(5,12\\right)[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 233.75px;\">[latex]\\left(12,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven that [latex]{h}^{-1}\\left(6\\right)=2[\/latex], what are the corresponding input and output values of the original function [latex]h?[\/latex]\r\n\r\n[reveal-answer q=\"664604\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"664604\"]\r\n\r\n[latex]h(2)=6[\/latex]\r\n\r\nThe input value to the original function [latex]h[\/latex] is [latex]x=2[\/latex] and the output value for [latex]h(2)[\/latex] is 6.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/TSztRfzmk0M\r\n\r\nhttps:\/\/youtu.be\/FIF8SdZkJc8?si=AxVnwl-PAcFzsj9b\r\n<h2>Verify that Functions are Inverses<\/h2>\r\nGiven a function [latex]f\\left(x\\right)[\/latex], we can verify whether some other function [latex]g\\left(x\\right)[\/latex] is the inverse of [latex]f\\left(x\\right)[\/latex] by checking whether both [latex]g\\left(f\\left(x\\right)\\right)=x[\/latex] and [latex]f\\left(g\\left(x\\right)\\right)=x[\/latex].\r\n\r\nFor example, [latex]y=4x[\/latex] and [latex]y=\\dfrac{1}{4}x[\/latex] are inverse functions.\r\n\r\n[latex]\\left({f}^{-1}\\circ f\\right)\\left(x\\right)={f}^{-1}\\left(f\\left(x\\right)\\right)={f}^{-1}\\left(4x\\right)=\\dfrac{1}{4}\\left(4x\\right)=x[\/latex]\r\n\r\nand\r\n\r\n[latex]\\left({f}^{}\\circ {f}^{-1}\\right)\\left(x\\right)=f\\left({f}^{-1}\\left(x\\right)\\right)=f\\left(\\dfrac{1}{4}x\\right)=4\\left(\\dfrac{1}{4}x\\right)=x[\/latex]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Verify that two functions [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] are inverses<\/h3>\r\n<ol>\r\n \t<li>Determine whether [latex]f\\left(g\\left(x\\right)\\right)=x[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)=x[\/latex].<\/li>\r\n \t<li>If both statements are true, then [latex]g={f}^{-1}[\/latex] and [latex]f={g}^{-1}[\/latex]. If either statement is false, then [latex]g\\ne {f}^{-1}[\/latex] and [latex]f\\ne {g}^{-1}[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Testing Inverse Relationships Algebraically<\/h3>\r\nIf [latex]f\\left(x\\right)=3x-2[\/latex] and [latex]g\\left(x\\right)=\\dfrac{x+2}{3}[\/latex], are [latex]g[\/latex] and [latex]f[\/latex] inverse functions?\r\n\r\n[reveal-answer q=\"421291\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"421291\"]\r\n\r\nFirst, we find [latex] g\\left(f\\left(x\\right)\\right)[\/latex].\r\n<p style=\"text-align: left;\">[latex]\\begin{align} g\\left(f\\left(x\\right)\\right)&amp;=\\dfrac{\\left(3x-2\\right)+2}{3}\\\\[1.5mm]&amp;=\\dfrac{3x-2+2}{3}\\\\[1.5mm]&amp;=\\dfrac{3x}{3}\\\\[1.5mm]&amp;={ x } \\end{align}[\/latex]<\/p>\r\nNext, find [latex] f\\left(g\\left(x\\right)\\right)[\/latex].\r\n<p style=\"text-align: left;\">[latex]\\begin{align} f\\left(g\\left(x\\right)\\right)&amp;=3\\left(\\dfrac{x+2}{3}\\right)-2\\\\[1.5mm]&amp;={ x }+{ 2 } -{ 2 }\\\\[1.5mm]&amp;={ x } \\end{align}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nAs [latex] g\\left(f\\left(x\\right)\\right)=x[\/latex] and [latex] f\\left(g\\left(x\\right)\\right)=x[\/latex], [latex]g[\/latex] and [latex]f[\/latex] are inverse functions.\r\n\r\nNotice the inverse operations are in reverse order of the operations from the original function: [latex]f\\left(x\\right)[\/latex] takes an input, first multiplies it by 3, then subtracts 2. The inverse function [latex]g\\left(x\\right)[\/latex] takes an input, adds 2, then divides by 3.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThese videos have more examples of algebraically verifying if two functions are inverse functions.\r\n\r\nhttps:\/\/youtu.be\/RLkrmkUaRYs?si=Oisyxh2s1zMMDADV\r\n\r\nhttps:\/\/youtu.be\/J9KpLnJYo5A?si=jj48LNFYel8aAHKg\r\n<h2>Finding Inverses of Linear Functions<\/h2>\r\nIf we have a linear function given as a formula\u2014for example, [latex]y[\/latex] as a function of [latex]x[\/latex]\u2014we can find the inverse function by solving to obtain [latex]x[\/latex] as a function of [latex]y[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Find the inverse of a linear function<\/h3>\r\n<ol>\r\n \t<li>Verify that\u00a0[latex]f[\/latex] is a one-to-one function.<\/li>\r\n \t<li>Replace [latex]f\\left(x\\right)[\/latex] with [latex]y[\/latex].<\/li>\r\n \t<li>Interchange [latex]x[\/latex]\u00a0and [latex]y[\/latex].<\/li>\r\n \t<li>Solve for [latex]y[\/latex], and rename the function [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\nThis video demonstrates finding the inverse of a linear function.\r\n\r\nhttps:\/\/youtu.be\/8nsaS4snzVY?si=cH9QyAK5Ufl9Q4nB\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Inverting the Fahrenheit-to-Celsius Function<\/h3>\r\nFind a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.\r\n<p style=\"text-align: left;\">[latex]C=\\frac{5}{9}\\left(F - 32\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"625400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"625400\"]\r\n<p style=\"text-align: left;\">[latex]\\begin{align}C&amp;=\\frac{5}{9}\\left(F - 32\\right) &amp;&amp; \\color{blue}{\\textsf{multiply both sides of the equation by $\\frac{9}{5}$}} \\\\[5pt]C\\cdot \\frac{9}{5}&amp;=F-32&amp;&amp; \\color{blue}{\\textsf{solve for $F$}} \\\\[5pt]F&amp;=\\frac{9}{5}C+32\\end{align}[\/latex]<\/p>\r\nBy solving in general, we have uncovered the inverse function. If\r\n<p style=\"text-align: left;\">[latex]C=h\\left(F\\right)=\\frac{5}{9}\\left(F - 32\\right)[\/latex],<\/p>\r\nthen\r\n<p style=\"text-align: left;\">[latex]F={h}^{-1}\\left(C\\right)=\\frac{9}{5}C+32[\/latex].<\/p>\r\nIn this case, we introduced a function [latex]h[\/latex] to represent the conversion because the input and output variables are descriptive, and writing [latex]{C}^{-1}[\/latex] could get confusing.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the inverse function of the one-to-one function [latex]f\\left(x\\right)=\\frac{1}{3}\\left(x - 5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"875458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875458\"]\r\n\r\n[latex]\\require{color}\\begin{align}\r\n\r\nf\\left(x\\right) &amp;= \\frac{1}{3}\\left(x - 5\\right)\\\\[5pt]\r\n\r\ny &amp;= \\frac{1}{3}\\left(x - 5\\right)&amp;&amp;\\\\[5pt]\r\n\r\nx &amp;= \\frac{1}{3}\\left(y - 5\\right)&amp;&amp; \\color{blue}{\\textsf{interchange $x$ and $y$}}\\\\[5pt]\r\n\r\n3\\left(x\\right) &amp;= 3\\left(\\frac{1}{3}\\left(y - 5\\right)\\right)&amp;&amp; \\color{blue}{\\textsf{solve for $y$}}\\\\[5pt]\r\n\r\n3x &amp;= y - 5\\\\[5pt]\r\n\r\n3x+5 &amp;= y\\\\[5pt]\r\n\r\ny &amp;= 3x+5&amp;&amp; \\color{blue}{\\textsf{rename the function ${f}^{-1}\\left(x\\right)$}}\\\\[5pt]\r\n\r\n{f}^{-1}\\left(x\\right)&amp;= 3x+5\\\\[5pt]\r\n\r\n\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Graph a Linear Function and its Inverse<\/h2>\r\nTo graph the inverse of a linear function, one approach is to find two or more points on the graph of the linear function. Then simply switch the [latex]x[\/latex]- and\u00a0[latex]y[\/latex]-coordinates of each point to find points that lie on the graph of the inverse function. Then use these points to graph the inverse function. For example, [latex](0, 0)[\/latex] and [latex](1, 2)[\/latex] are two points that lie on the graph of the function [latex]f(x)=2x[\/latex]. To graph the inverse function, switch the [latex]x[\/latex]- and [latex]y[\/latex]-coordinates of these points to get [latex](0, 0)[\/latex] and [latex](2, 1)[\/latex]. The graph of the inverse function will be the line that passes through the two points [latex](0, 0)[\/latex] and [latex](2, 1)[\/latex] (See below).\r\n\r\n[caption id=\"attachment_1068\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1068 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/10232610\/2-5-1-InverseGraph-2-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/> Graph of the function [latex]f(x)=2x[\/latex] and its inverse.[\/caption]\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nUse the graph of the linear function [latex]f(x)=4x-2[\/latex] to graph the inverse function.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1392\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1392\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014041\/fx4x-2-300x298.png\" alt=\"Graph of f(x)=4x-2\" width=\"300\" height=\"298\" \/> Graph of [latex]f(x)=4x-2[\/latex][\/caption][reveal-answer q=\"420085\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"420085\"]Choose 2 (or more) points on the original line: [latex](0, \u20132)[\/latex] and [latex](1, 2)[\/latex].To graph the inverse linear function, reverse the [latex]x[\/latex]- and [latex]y[\/latex]-coordinates: [latex](\u20132, 0)[\/latex] and [latex](2, 1)[\/latex].Plot the new points. The line that passes through them is the graph of the inverse function.[caption id=\"attachment_1391\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1391\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07013646\/fx4x-2-and-inverse-300x294.png\" alt=\"Graph of f(x)=4x-2 and its inverse\" width=\"300\" height=\"294\" \/> Graph of [latex]f(x)=4x-2[\/latex] and its inverse[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nUse the graphs of the linear functions to graph their inverse functions.\r\n<table border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td>1.\r\n\r\n[caption id=\"attachment_1393\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1393\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014325\/fx-3x5-300x297.png\" alt=\"Graph of f(x)=-3x+5\" width=\"300\" height=\"297\" \/> Graph of [latex]f(x)=-3x+5[\/latex][\/caption]<\/td>\r\n<td>1.\r\n\r\n[caption id=\"attachment_1397\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1397\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07020047\/fx43x2-300x294.png\" alt=\"Graph of f(x)=4\/3 x+2\" width=\"300\" height=\"294\" \/> Graph of [latex]f(x)=\\frac{4}{3}x+2[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"hjm853\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm853\"]\r\n<table border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td>1.\r\n\r\n[caption id=\"attachment_1394\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1394\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014620\/fx-3x5-and-inverse-300x300.png\" alt=\"Graph of f(x)=-3x+5 and its inverse\" width=\"300\" height=\"300\" \/> Graph of [latex]f(x)=-3x+5[\/latex] and its inverse[\/caption]<\/td>\r\n<td>2.\r\n\r\n[caption id=\"attachment_1398\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1398\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07020052\/fx43x2-and-inverse-300x297.png\" alt=\"Graph of f(x)=4\/3 x+2 and its inverse\" width=\"300\" height=\"297\" \/> Graph of [latex]f(x)=\\frac{4}{3}x+2[\/latex] and its inverse[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFind the inverse of the function [latex]f(x)=\\frac{2}{5}x+2[\/latex], then graph the function and its inverse.\r\n\r\n[reveal-answer q=\"149434\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"149434\"]\r\n\r\nFirst find the inverse function.\r\n\r\n[latex]\\require{color}\\begin{align}\r\n\r\nf\\left(x\\right) &amp;= \\frac{2}{5}x+2&amp;&amp; \\color{blue}{\\textsf{write with $y$}}\\\\[5pt]\r\n\r\ny &amp;= \\frac{2}{5}x+2&amp;&amp; \\color{blue}{\\textsf{interchange $x$ and $y$}}\\\\[5pt]\r\n\r\nx &amp;= \\frac{2}{5}y+2&amp;&amp; \\color{blue}{\\textsf{multiply both sides of the equation by $5$}}\\\\[5pt]\r\n\r\n5\\left(x\\right) &amp;= 5\\left(\\frac{2}{5}y+2\\right)&amp;&amp; \\color{blue}{\\textsf{solve for $y$}}\\\\[5pt]\r\n\r\n5x &amp;= 2y+10\\\\[5pt]\r\n\r\n5x-10 &amp;= 2y\\\\[5pt]\r\n\r\n\\frac{5x-10}{2}&amp;=y\\\\[5pt]\r\n\r\ny &amp;= \\frac{5}{2}x-5&amp;&amp; \\color{blue}{\\textsf{rename the function ${f}^{-1}\\left(x\\right)$}}\\\\[5pt]\r\n\r\n{f}^{-1}\\left(x\\right)&amp;= \\frac{5}{2}x-5\\\\[5pt]\r\n\r\n\\end{align}[\/latex]\r\n\r\n&nbsp;\r\n\r\nNext, graph the function and its inverse. To graph\u00a0[latex]f(x)=\\frac{2}{5}x+2[\/latex] we can either use a table of values or use the slope and [latex]y[\/latex]-intercept.\r\n\r\nThe slope of the line is [latex]m=\\frac{2}{5}[\/latex] and the [latex]y[\/latex]-intercept is [latex](0, 2)[\/latex].\r\n\r\nTo graph the line, we start at [latex](0, 2)[\/latex] then run [latex]5[\/latex] units to the right and [latex]2[\/latex] units up to get to [latex](5, 4)[\/latex].\r\n\r\n[caption id=\"attachment_1400\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1400\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07021055\/fx25-x2-300x291.png\" alt=\"Graph of f(x)=2\/5 x+2\" width=\"300\" height=\"291\" \/> Graph of [latex]f(x)=\\frac{2}{5}x+2[\/latex][\/caption]To graph\u00a0[latex]f^{-1}\\left(x\\right)=\\frac{5}{2}x-5[\/latex] we can use a table of values, use the slope and [latex]y[\/latex]-intercept, or reverse some of the points from [latex]f(x)[\/latex].Let's use the slope and intercept again. The slope of the line is [latex]m=\\frac{5}{2}[\/latex] and the [latex]y[\/latex]-intercept is [latex](0, -5)[\/latex].To graph the line, we start at [latex](0, -5)[\/latex] then run [latex]2[\/latex] units to the right and [latex]5[\/latex] units up to get to [latex](2,0)[\/latex].[caption id=\"attachment_1399\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1399\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07021050\/fx25-x2-and-inverse-300x297.png\" alt=\"Graph of f(x)=2\/5 x+2 and its inverse\" width=\"300\" height=\"297\" \/> Graph of [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse [latex]f^{-1}(x)=\\frac{5}{2}x-5[\/latex][\/caption]Notice that if we instead reversed the points [latex](0, 2)[\/latex] and [latex](5, 4)[\/latex] from [latex]f(x)[\/latex] to [latex](2, 0)[\/latex] and [latex](4, 5)[\/latex], we could draw the same line to graph the inverse function through these new points.Notice the relationship of the slope of the original function to the slope of the inverse function. The original function has a slope of [latex]\\frac{2}{5}[\/latex], while the inverse function has a slope of [latex]\\frac{5}{2}[\/latex]. i.e. the slopes are reciprocals of each other, but NOT negative reciprocals like perpendicular lines.[\/hidden-answer]<\/div>\r\n<h2>Inverse Functions and Symmetry<\/h2>\r\nThe graph of any function and its inverse are symmetric across the line [latex]y=x[\/latex].\u00a0 For example, the graph below shows the graph of [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse, [latex]f^{-1}(x)=\\frac{5}{2}x-5[\/latex]. Adding the dashed line [latex]y=x[\/latex], we can see that the two lines are symmetric (mirror images of one another) across the line [latex]y=x[\/latex].\r\n\r\n[caption id=\"attachment_1409\" align=\"aligncenter\" width=\"318\"]<img class=\"wp-image-1409\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07033342\/Inverse-functions-and-symmetry-300x293.png\" alt=\"Graph showing symmetry of inverse functions across the line y=x\" width=\"318\" height=\"310\" \/> Graph of [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse [latex]f^{-1}(x)=\\frac{5}{2}x-5[\/latex][\/caption]Notice the [latex]x[\/latex]- and [latex]y[\/latex]-intercepts. The [latex]y[\/latex]-intercept of [latex](0, 2)[\/latex] in the original function (blue line) reflects to the [latex]x[\/latex]-intercept [latex](2, 0)[\/latex] in the inverse function (green line). Also, the\u00a0[latex]x[\/latex]-intercept of [latex](\u20135, 0)[\/latex] in the original function (blue line) reflects to the [latex]y[\/latex]-intercept [latex](0, \u20135)[\/latex] in the inverse function (green line).\r\n\r\nThe slopes of each function are also related. The function [latex]f(x)[\/latex] has a slope of [latex]\\frac{2}{5}[\/latex], while the inverse function [latex]f^{-1}(x)[\/latex] has a slope of [latex]\\frac{5}{2}[\/latex]. <strong>The slopes of inverse functions are reciprocals of each other. <\/strong>This is\u00a0because the slope of a function is [latex]\\dfrac{\\text{change in y}}{\\text{change in x}}[\/latex]. The slope of the inverse function becomes\u00a0[latex]\\dfrac{\\text{change in x}}{\\text{change in y}}[\/latex].","rendered":"<div class=\"textbox learning-objectives\">\n<h3 style=\"text-align: center;\">Learning Outcomes<\/h3>\n<ul>\n<li>Given a table or graph, determine if a function is one-to-one.<\/li>\n<li>Find the inverse of a linear function.<\/li>\n<li>Determine if two functions are inverses by finding [latex]f(f^{-1}(x))[\/latex] and [latex]f^{-1}(f(x))[\/latex].<\/li>\n<li>Graph a linear function and its inverse on the same axes.<\/li>\n<\/ul>\n<\/div>\n<div class=\"mceTemp\"><\/div>\n<h2>One-to-One Functions<\/h2>\n<p>Recall that in a function, the input value must have one and only one value for the output.\u00a0The set of input values is the <strong>domain of the function<\/strong> and the set of output values is the <strong>range of the function<\/strong>.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the domain and range for the function.<\/p>\n<table style=\"height: 134px; width: 348px; width: 409px;\">\n<thead>\n<tr>\n<th style=\"text-align: center; width: 257.683px;\"><strong><i>[latex]x[\/latex]<\/i><\/strong><\/th>\n<th style=\"text-align: center; width: 224.95px;\"><strong><i>[latex]y[\/latex]<\/i><\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center; width: 257.683px;\">[latex]\u22125[\/latex]<\/td>\n<td style=\"text-align: center; width: 224.95px;\">[latex]\u22126[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 257.683px;\">[latex]\u22122[\/latex]<\/td>\n<td style=\"text-align: center; width: 224.95px;\">[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 257.683px;\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center; width: 224.95px;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 257.683px;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center; width: 224.95px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 257.683px;\">[latex]5[\/latex]<\/td>\n<td style=\"text-align: center; width: 224.95px;\">[latex]15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q130987\">Show Solution<\/span><\/p>\n<div id=\"q130987\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is the set of inputs or [latex]x[\/latex]-coordinates.<\/p>\n<p style=\"text-align: center;\">[latex]\\{\u22125,\u22122,\u22121,0,5\\}[\/latex]<\/p>\n<p>The range is the set of outputs or\u00a0[latex]y[\/latex]-coordinates.<\/p>\n<p style=\"text-align: center;\">[latex]\\{\u22126,\u22121,0,3,15\\}[\/latex]<\/p>\n<p style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<p>Some functions have a given output value that corresponds to two or more input values. For example, in the following stock chart the stock price was\u00a0[latex]$1000[\/latex] on five different dates, meaning that there were five different input values that all resulted in the same output value of\u00a0[latex]$1000[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200506\/CNX_Precalc_Figure_01_00_001n2.jpg\" alt=\"Figure of a bull and a graph of market prices.\" width=\"975\" height=\"307\" \/><\/p>\n<p id=\"fs-id1165135678633\">However, some functions have only one input value for each output value as well as having only one output value for each input value. We call these functions <strong>one-to-one functions<\/strong>. As an example, consider a school that uses only letter grades and decimal equivalents as listed below.<\/p>\n<table style=\"width: 353px;\" summary=\"Two columns and five rows. The first column is labeled,\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: center; width: 234.55px;\">Letter grade<\/th>\n<th style=\"text-align: center; width: 278.717px;\">Grade point average<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center; width: 234.55px;\">A<\/td>\n<td style=\"text-align: center; width: 278.717px;\">[latex]4.0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 234.55px;\">B<\/td>\n<td style=\"text-align: center; width: 278.717px;\">[latex]3.0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 234.55px;\">C<\/td>\n<td style=\"text-align: center; width: 278.717px;\">[latex]2.0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 234.55px;\">D<\/td>\n<td style=\"text-align: center; width: 278.717px;\">[latex]1.0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137561844\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\n<p>To visualize this concept, look at the two simple functions sketched in (a) and (b) below. Note that (c) is not a function since the input\u00a0<em>q<\/em> produces two outputs,\u00a0<em>y<\/em> and\u00a0<em>z<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200453\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/><\/p>\n<p>The function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>One-to-One Function<\/h3>\n<p>A one-to-one function is a function in which each output value corresponds to exactly one input value.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine whether each of the following tables represents a one-to-one function.<\/p>\n<p>a)<\/p>\n<table style=\"width: 234px;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; width: 235.45px;\"><strong>Input<\/strong><\/td>\n<td style=\"text-align: center; width: 247.183px;\"><strong>Output<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 235.45px;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center; width: 247.183px;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 235.45px;\">[latex]12[\/latex]<\/td>\n<td style=\"text-align: center; width: 247.183px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 235.45px;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center; width: 247.183px;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 235.45px;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center; width: 247.183px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 235.45px;\">[latex]-5[\/latex]<\/td>\n<td style=\"text-align: center; width: 247.183px;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>b)<\/p>\n<table style=\"width: 234px;\">\n<tbody>\n<tr>\n<td style=\"text-align: center; width: 237.367px;\"><strong>Input<\/strong><\/td>\n<td style=\"text-align: center; width: 245.267px;\"><strong>Output<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 237.367px;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center; width: 245.267px;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 237.367px;\">[latex]8[\/latex]<\/td>\n<td style=\"text-align: center; width: 245.267px;\">[latex]16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 237.367px;\">[latex]16[\/latex]<\/td>\n<td style=\"text-align: center; width: 245.267px;\">[latex]32[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 237.367px;\">[latex]32[\/latex]<\/td>\n<td style=\"text-align: center; width: 245.267px;\">[latex]64[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 237.367px;\">[latex]64[\/latex]<\/td>\n<td style=\"text-align: center; width: 245.267px;\">[latex]128[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q945171\">Show Solution<\/span><\/p>\n<div id=\"q945171\" class=\"hidden-answer\" style=\"display: none\">\n<p>Table a) maps the output value\u00a0[latex]2[\/latex] to two different input values, therefore\u00a0this is NOT a one-to-one function.<\/p>\n<p>Table b) maps each output to one unique input, therefore this IS a one-to-one function.<\/p>\n<p>Only table b) is one-to-one.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show an example of using tables of values to determine whether a function is one-to-one.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Determine if a Relation Given as a Table is a One-to-One Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QFOJmevha_Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using the Horizontal Line Test<\/h2>\n<p id=\"fs-id1165137871503\">An easy way to determine whether a function\u00a0is a one-to-one function is to use the <strong>horizontal line test <\/strong>on the graph of the function. \u00a0To do this, draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>For\u00a0the following graphs, determine which represent one-to-one functions.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200511\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783411\">Show Solution<\/span><\/p>\n<div id=\"q783411\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135185190\">The function in (a) is\u00a0not one-to-one. Using the horizontal line test, as shown below, it intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200515\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"\" width=\"294\" height=\"267\" \/><\/p>\n<figure id=\"Figure_01_01_010\" class=\"small\"><\/figure>\n<p>The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2697 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212200\/Screen-Shot-2016-07-16-at-2.21.48-PM-287x300.png\" alt=\"Graph of a line with three dashed horizontal lines passing through it.\" width=\"287\" height=\"300\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>The function (c) is not one-to-one and is in fact not a function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2698 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212527\/Screen-Shot-2016-07-16-at-2.25.36-PM-237x300.png\" alt=\"Graph of a circle with two dashed lines passing through horizontally\" width=\"279\" height=\"353\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of using the horizontal line test to determine whether a graph represents a one-to-one function.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Determine if the Graph of a Relation is a One-to-One Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tbSGdcSN8RE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Inverse Functions<\/h2>\n<p>Given a one-to-one function [latex]f\\left(x\\right)[\/latex], we represent its inverse as [latex]{f}^{-1}\\left(x\\right)[\/latex], read as &#8220;[latex]f[\/latex] inverse of [latex]x[\/latex].&#8221; The superscript [latex]-1[\/latex] is part of the notation. It is not an exponent; it does not imply a power of [latex]-1[\/latex] . In other words, [latex]{f}^{-1}\\left(x\\right)[\/latex] does <em>not<\/em> mean [latex]\\dfrac{1}{f\\left(x\\right)}[\/latex]. Instead, [latex]\\dfrac{1}{f\\left(x\\right)}[\/latex] is the reciprocal of [latex]f[\/latex], not the inverse.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Inverse Function<\/h3>\n<p>For any <strong>one-to-one function<\/strong> [latex]f\\left(x\\right)=y[\/latex], a function [latex]{f}^{-1}\\left(x\\right)[\/latex] is an <strong>inverse function<\/strong> of [latex]f[\/latex] if [latex]{f}^{-1}\\left(y\\right)=x[\/latex]. This can also be written as [latex]{f}^{-1}\\left(f\\left(x\\right)\\right)=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]. It also follows that [latex]f\\left({f}^{-1}\\left(x\\right)\\right)=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]{f}^{-1}[\/latex] if [latex]{f}^{-1}[\/latex] is the inverse of [latex]f[\/latex].<\/p>\n<p>The notation [latex]{f}^{-1}[\/latex] is read &#8220;[latex]f[\/latex] inverse.&#8221; Like any other function, we can use any variable name as the input for [latex]{f}^{-1}[\/latex], so we will often write [latex]{f}^{-1}\\left(x\\right)[\/latex], which we read as [latex]\"f[\/latex] inverse of [latex]x[\/latex]&#8220;.<\/p>\n<\/div>\n<h4 style=\"padding-right: 120px;\">Why -1?<\/h4>\n<p style=\"padding-right: 120px;\">The &#8220;exponent-like&#8221; notation comes from an analogy between function composition and multiplication: just as [latex]{a}^{-1}a=1[\/latex] for any nonzero number [latex]a[\/latex], so [latex]\\left({f}^{-1}\\circ f\\right)\\left(x\\right)={f}^{-1}\\left(f\\left(x\\right)\\right)={f}^{-1}\\left(y\\right)=x[\/latex]. This is true for all [latex]x[\/latex] in the domain of [latex]f[\/latex]. Informally, this means that inverse functions &#8220;undo&#8221; each other. However, just as zero does not have a <strong>reciprocal<\/strong>, some functions do not have inverses.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying an Inverse Function for a Given Input-Output Pair<\/h3>\n<p>If for a particular one-to-one function [latex]f\\left(2\\right)=4[\/latex] and [latex]f\\left(5\\right)=12[\/latex], what are the corresponding input and output values for the inverse function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q348517\">Show Solution<\/span><\/p>\n<div id=\"q348517\" class=\"hidden-answer\" style=\"display: none\">\n<p>The inverse function reverses the input and output quantities, so if<\/p>\n<p style=\"text-align: left;\">[latex]f\\left(2\\right)=4[\/latex], then [latex]{f}^{-1}\\left(4\\right)=2[\/latex]<\/p>\n<p style=\"text-align: left;\">and if<\/p>\n<p style=\"text-align: left;\">[latex]f\\left(5\\right)=12[\/latex], then [latex]{f}^{-1}\\left(12\\right)=5[\/latex]<\/p>\n<p>Alternatively, if we want to name the inverse function [latex]g[\/latex], then [latex]g\\left(4\\right)=2[\/latex] and [latex]g\\left(12\\right)=5[\/latex].<\/p>\n<p>If we show the coordinate pairs in a table form, it is clear that the the inputs and outputs are reversed.<\/p>\n<table style=\"width: 315px;\" summary=\"For (x,f(x)) we have the values (2, 4) and (5, 12); for (x, g(x)), we have the values (4, 2) and (12, 5).\">\n<thead>\n<tr>\n<th style=\"text-align: center; width: 248.883px;\">[latex]\\left(x,f\\left(x\\right)\\right)[\/latex]<\/th>\n<th style=\"text-align: center; width: 233.75px;\">[latex]\\left(x,g\\left(x\\right)\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center; width: 248.883px;\">[latex]\\left(2,4\\right)[\/latex]<\/td>\n<td style=\"text-align: center; width: 233.75px;\">[latex]\\left(4,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; width: 248.883px;\">[latex]\\left(5,12\\right)[\/latex]<\/td>\n<td style=\"text-align: center; width: 233.75px;\">[latex]\\left(12,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given that [latex]{h}^{-1}\\left(6\\right)=2[\/latex], what are the corresponding input and output values of the original function [latex]h?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q664604\">Show Solution<\/span><\/p>\n<div id=\"q664604\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]h(2)=6[\/latex]<\/p>\n<p>The input value to the original function [latex]h[\/latex] is [latex]x=2[\/latex] and the output value for [latex]h(2)[\/latex] is 6.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Find an Inverse Function From a Table\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/TSztRfzmk0M?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex:  Function and Inverse Function Values Using a Graph\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/FIF8SdZkJc8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Verify that Functions are Inverses<\/h2>\n<p>Given a function [latex]f\\left(x\\right)[\/latex], we can verify whether some other function [latex]g\\left(x\\right)[\/latex] is the inverse of [latex]f\\left(x\\right)[\/latex] by checking whether both [latex]g\\left(f\\left(x\\right)\\right)=x[\/latex] and [latex]f\\left(g\\left(x\\right)\\right)=x[\/latex].<\/p>\n<p>For example, [latex]y=4x[\/latex] and [latex]y=\\dfrac{1}{4}x[\/latex] are inverse functions.<\/p>\n<p>[latex]\\left({f}^{-1}\\circ f\\right)\\left(x\\right)={f}^{-1}\\left(f\\left(x\\right)\\right)={f}^{-1}\\left(4x\\right)=\\dfrac{1}{4}\\left(4x\\right)=x[\/latex]<\/p>\n<p>and<\/p>\n<p>[latex]\\left({f}^{}\\circ {f}^{-1}\\right)\\left(x\\right)=f\\left({f}^{-1}\\left(x\\right)\\right)=f\\left(\\dfrac{1}{4}x\\right)=4\\left(\\dfrac{1}{4}x\\right)=x[\/latex]<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Verify that two functions [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] are inverses<\/h3>\n<ol>\n<li>Determine whether [latex]f\\left(g\\left(x\\right)\\right)=x[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)=x[\/latex].<\/li>\n<li>If both statements are true, then [latex]g={f}^{-1}[\/latex] and [latex]f={g}^{-1}[\/latex]. If either statement is false, then [latex]g\\ne {f}^{-1}[\/latex] and [latex]f\\ne {g}^{-1}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Testing Inverse Relationships Algebraically<\/h3>\n<p>If [latex]f\\left(x\\right)=3x-2[\/latex] and [latex]g\\left(x\\right)=\\dfrac{x+2}{3}[\/latex], are [latex]g[\/latex] and [latex]f[\/latex] inverse functions?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q421291\">Show Solution<\/span><\/p>\n<div id=\"q421291\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we find [latex]g\\left(f\\left(x\\right)\\right)[\/latex].<\/p>\n<p style=\"text-align: left;\">[latex]\\begin{align} g\\left(f\\left(x\\right)\\right)&=\\dfrac{\\left(3x-2\\right)+2}{3}\\\\[1.5mm]&=\\dfrac{3x-2+2}{3}\\\\[1.5mm]&=\\dfrac{3x}{3}\\\\[1.5mm]&={ x } \\end{align}[\/latex]<\/p>\n<p>Next, find [latex]f\\left(g\\left(x\\right)\\right)[\/latex].<\/p>\n<p style=\"text-align: left;\">[latex]\\begin{align} f\\left(g\\left(x\\right)\\right)&=3\\left(\\dfrac{x+2}{3}\\right)-2\\\\[1.5mm]&={ x }+{ 2 } -{ 2 }\\\\[1.5mm]&={ x } \\end{align}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>As [latex]g\\left(f\\left(x\\right)\\right)=x[\/latex] and [latex]f\\left(g\\left(x\\right)\\right)=x[\/latex], [latex]g[\/latex] and [latex]f[\/latex] are inverse functions.<\/p>\n<p>Notice the inverse operations are in reverse order of the operations from the original function: [latex]f\\left(x\\right)[\/latex] takes an input, first multiplies it by 3, then subtracts 2. The inverse function [latex]g\\left(x\\right)[\/latex] takes an input, adds 2, then divides by 3.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>These videos have more examples of algebraically verifying if two functions are inverse functions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Determine if Two Linear Functions Are Inverses (1)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/RLkrmkUaRYs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Determine if Two Linear Functions Are Inverses (2)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/J9KpLnJYo5A?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Finding Inverses of Linear Functions<\/h2>\n<p>If we have a linear function given as a formula\u2014for example, [latex]y[\/latex] as a function of [latex]x[\/latex]\u2014we can find the inverse function by solving to obtain [latex]x[\/latex] as a function of [latex]y[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Find the inverse of a linear function<\/h3>\n<ol>\n<li>Verify that\u00a0[latex]f[\/latex] is a one-to-one function.<\/li>\n<li>Replace [latex]f\\left(x\\right)[\/latex] with [latex]y[\/latex].<\/li>\n<li>Interchange [latex]x[\/latex]\u00a0and [latex]y[\/latex].<\/li>\n<li>Solve for [latex]y[\/latex], and rename the function [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<p>This video demonstrates finding the inverse of a linear function.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Find an Inverse Function Value Given a Linear Function #math #maths #mathinstruction\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8nsaS4snzVY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Inverting the Fahrenheit-to-Celsius Function<\/h3>\n<p>Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.<\/p>\n<p style=\"text-align: left;\">[latex]C=\\frac{5}{9}\\left(F - 32\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q625400\">Show Solution<\/span><\/p>\n<div id=\"q625400\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex]\\begin{align}C&=\\frac{5}{9}\\left(F - 32\\right) && \\color{blue}{\\textsf{multiply both sides of the equation by $\\frac{9}{5}$}} \\\\[5pt]C\\cdot \\frac{9}{5}&=F-32&& \\color{blue}{\\textsf{solve for $F$}} \\\\[5pt]F&=\\frac{9}{5}C+32\\end{align}[\/latex]<\/p>\n<p>By solving in general, we have uncovered the inverse function. If<\/p>\n<p style=\"text-align: left;\">[latex]C=h\\left(F\\right)=\\frac{5}{9}\\left(F - 32\\right)[\/latex],<\/p>\n<p>then<\/p>\n<p style=\"text-align: left;\">[latex]F={h}^{-1}\\left(C\\right)=\\frac{9}{5}C+32[\/latex].<\/p>\n<p>In this case, we introduced a function [latex]h[\/latex] to represent the conversion because the input and output variables are descriptive, and writing [latex]{C}^{-1}[\/latex] could get confusing.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the inverse function of the one-to-one function [latex]f\\left(x\\right)=\\frac{1}{3}\\left(x - 5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q875458\">Show Solution<\/span><\/p>\n<div id=\"q875458\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\require{color}\\begin{align}    f\\left(x\\right) &= \\frac{1}{3}\\left(x - 5\\right)\\\\[5pt]    y &= \\frac{1}{3}\\left(x - 5\\right)&&\\\\[5pt]    x &= \\frac{1}{3}\\left(y - 5\\right)&& \\color{blue}{\\textsf{interchange $x$ and $y$}}\\\\[5pt]    3\\left(x\\right) &= 3\\left(\\frac{1}{3}\\left(y - 5\\right)\\right)&& \\color{blue}{\\textsf{solve for $y$}}\\\\[5pt]    3x &= y - 5\\\\[5pt]    3x+5 &= y\\\\[5pt]    y &= 3x+5&& \\color{blue}{\\textsf{rename the function ${f}^{-1}\\left(x\\right)$}}\\\\[5pt]    {f}^{-1}\\left(x\\right)&= 3x+5\\\\[5pt]    \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Graph a Linear Function and its Inverse<\/h2>\n<p>To graph the inverse of a linear function, one approach is to find two or more points on the graph of the linear function. Then simply switch the [latex]x[\/latex]&#8211; and\u00a0[latex]y[\/latex]-coordinates of each point to find points that lie on the graph of the inverse function. Then use these points to graph the inverse function. For example, [latex](0, 0)[\/latex] and [latex](1, 2)[\/latex] are two points that lie on the graph of the function [latex]f(x)=2x[\/latex]. To graph the inverse function, switch the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates of these points to get [latex](0, 0)[\/latex] and [latex](2, 1)[\/latex]. The graph of the inverse function will be the line that passes through the two points [latex](0, 0)[\/latex] and [latex](2, 1)[\/latex] (See below).<\/p>\n<div id=\"attachment_1068\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1068\" class=\"wp-image-1068 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/10232610\/2-5-1-InverseGraph-2-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-1068\" class=\"wp-caption-text\">Graph of the function [latex]f(x)=2x[\/latex] and its inverse.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Use the graph of the linear function [latex]f(x)=4x-2[\/latex] to graph the inverse function.<\/p>\n<p>&nbsp;<\/p>\n<div id=\"attachment_1392\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1392\" class=\"wp-image-1392\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014041\/fx4x-2-300x298.png\" alt=\"Graph of f(x)=4x-2\" width=\"300\" height=\"298\" \/><\/p>\n<p id=\"caption-attachment-1392\" class=\"wp-caption-text\">Graph of [latex]f(x)=4x-2[\/latex]<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q420085\">Show Answer<\/span><\/p>\n<div id=\"q420085\" class=\"hidden-answer\" style=\"display: none\">Choose 2 (or more) points on the original line: [latex](0, \u20132)[\/latex] and [latex](1, 2)[\/latex].To graph the inverse linear function, reverse the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates: [latex](\u20132, 0)[\/latex] and [latex](2, 1)[\/latex].Plot the new points. The line that passes through them is the graph of the inverse function.<\/p>\n<div id=\"attachment_1391\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1391\" class=\"wp-image-1391\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07013646\/fx4x-2-and-inverse-300x294.png\" alt=\"Graph of f(x)=4x-2 and its inverse\" width=\"300\" height=\"294\" \/><\/p>\n<p id=\"caption-attachment-1391\" class=\"wp-caption-text\">Graph of [latex]f(x)=4x-2[\/latex] and its inverse<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Use the graphs of the linear functions to graph their inverse functions.<\/p>\n<table>\n<tbody>\n<tr>\n<td>1.<\/p>\n<div id=\"attachment_1393\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1393\" class=\"wp-image-1393\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014325\/fx-3x5-300x297.png\" alt=\"Graph of f(x)=-3x+5\" width=\"300\" height=\"297\" \/><\/p>\n<p id=\"caption-attachment-1393\" class=\"wp-caption-text\">Graph of [latex]f(x)=-3x+5[\/latex]<\/p>\n<\/div>\n<\/td>\n<td>1.<\/p>\n<div id=\"attachment_1397\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1397\" class=\"wp-image-1397\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07020047\/fx43x2-300x294.png\" alt=\"Graph of f(x)=4\/3 x+2\" width=\"300\" height=\"294\" \/><\/p>\n<p id=\"caption-attachment-1397\" class=\"wp-caption-text\">Graph of [latex]f(x)=\\frac{4}{3}x+2[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm853\">Show Answer<\/span><\/p>\n<div id=\"qhjm853\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>1.<\/p>\n<div id=\"attachment_1394\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1394\" class=\"wp-image-1394\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014620\/fx-3x5-and-inverse-300x300.png\" alt=\"Graph of f(x)=-3x+5 and its inverse\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-1394\" class=\"wp-caption-text\">Graph of [latex]f(x)=-3x+5[\/latex] and its inverse<\/p>\n<\/div>\n<\/td>\n<td>2.<\/p>\n<div id=\"attachment_1398\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1398\" class=\"wp-image-1398\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07020052\/fx43x2-and-inverse-300x297.png\" alt=\"Graph of f(x)=4\/3 x+2 and its inverse\" width=\"300\" height=\"297\" \/><\/p>\n<p id=\"caption-attachment-1398\" class=\"wp-caption-text\">Graph of [latex]f(x)=\\frac{4}{3}x+2[\/latex] and its inverse<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Find the inverse of the function [latex]f(x)=\\frac{2}{5}x+2[\/latex], then graph the function and its inverse.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149434\">Show Answer<\/span><\/p>\n<div id=\"q149434\" class=\"hidden-answer\" style=\"display: none\">\n<p>First find the inverse function.<\/p>\n<p>[latex]\\require{color}\\begin{align}    f\\left(x\\right) &= \\frac{2}{5}x+2&& \\color{blue}{\\textsf{write with $y$}}\\\\[5pt]    y &= \\frac{2}{5}x+2&& \\color{blue}{\\textsf{interchange $x$ and $y$}}\\\\[5pt]    x &= \\frac{2}{5}y+2&& \\color{blue}{\\textsf{multiply both sides of the equation by $5$}}\\\\[5pt]    5\\left(x\\right) &= 5\\left(\\frac{2}{5}y+2\\right)&& \\color{blue}{\\textsf{solve for $y$}}\\\\[5pt]    5x &= 2y+10\\\\[5pt]    5x-10 &= 2y\\\\[5pt]    \\frac{5x-10}{2}&=y\\\\[5pt]    y &= \\frac{5}{2}x-5&& \\color{blue}{\\textsf{rename the function ${f}^{-1}\\left(x\\right)$}}\\\\[5pt]    {f}^{-1}\\left(x\\right)&= \\frac{5}{2}x-5\\\\[5pt]    \\end{align}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Next, graph the function and its inverse. To graph\u00a0[latex]f(x)=\\frac{2}{5}x+2[\/latex] we can either use a table of values or use the slope and [latex]y[\/latex]-intercept.<\/p>\n<p>The slope of the line is [latex]m=\\frac{2}{5}[\/latex] and the [latex]y[\/latex]-intercept is [latex](0, 2)[\/latex].<\/p>\n<p>To graph the line, we start at [latex](0, 2)[\/latex] then run [latex]5[\/latex] units to the right and [latex]2[\/latex] units up to get to [latex](5, 4)[\/latex].<\/p>\n<div id=\"attachment_1400\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1400\" class=\"wp-image-1400\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07021055\/fx25-x2-300x291.png\" alt=\"Graph of f(x)=2\/5 x+2\" width=\"300\" height=\"291\" \/><\/p>\n<p id=\"caption-attachment-1400\" class=\"wp-caption-text\">Graph of [latex]f(x)=\\frac{2}{5}x+2[\/latex]<\/p>\n<\/div>\n<p>To graph\u00a0[latex]f^{-1}\\left(x\\right)=\\frac{5}{2}x-5[\/latex] we can use a table of values, use the slope and [latex]y[\/latex]-intercept, or reverse some of the points from [latex]f(x)[\/latex].Let&#8217;s use the slope and intercept again. The slope of the line is [latex]m=\\frac{5}{2}[\/latex] and the [latex]y[\/latex]-intercept is [latex](0, -5)[\/latex].To graph the line, we start at [latex](0, -5)[\/latex] then run [latex]2[\/latex] units to the right and [latex]5[\/latex] units up to get to [latex](2,0)[\/latex].<\/p>\n<div id=\"attachment_1399\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1399\" class=\"wp-image-1399\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07021050\/fx25-x2-and-inverse-300x297.png\" alt=\"Graph of f(x)=2\/5 x+2 and its inverse\" width=\"300\" height=\"297\" \/><\/p>\n<p id=\"caption-attachment-1399\" class=\"wp-caption-text\">Graph of [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse [latex]f^{-1}(x)=\\frac{5}{2}x-5[\/latex]<\/p>\n<\/div>\n<p>Notice that if we instead reversed the points [latex](0, 2)[\/latex] and [latex](5, 4)[\/latex] from [latex]f(x)[\/latex] to [latex](2, 0)[\/latex] and [latex](4, 5)[\/latex], we could draw the same line to graph the inverse function through these new points.Notice the relationship of the slope of the original function to the slope of the inverse function. The original function has a slope of [latex]\\frac{2}{5}[\/latex], while the inverse function has a slope of [latex]\\frac{5}{2}[\/latex]. i.e. the slopes are reciprocals of each other, but NOT negative reciprocals like perpendicular lines.<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Inverse Functions and Symmetry<\/h2>\n<p>The graph of any function and its inverse are symmetric across the line [latex]y=x[\/latex].\u00a0 For example, the graph below shows the graph of [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse, [latex]f^{-1}(x)=\\frac{5}{2}x-5[\/latex]. Adding the dashed line [latex]y=x[\/latex], we can see that the two lines are symmetric (mirror images of one another) across the line [latex]y=x[\/latex].<\/p>\n<div id=\"attachment_1409\" style=\"width: 328px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1409\" class=\"wp-image-1409\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07033342\/Inverse-functions-and-symmetry-300x293.png\" alt=\"Graph showing symmetry of inverse functions across the line y=x\" width=\"318\" height=\"310\" \/><\/p>\n<p id=\"caption-attachment-1409\" class=\"wp-caption-text\">Graph of [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse [latex]f^{-1}(x)=\\frac{5}{2}x-5[\/latex]<\/p>\n<\/div>\n<p>Notice the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-intercepts. The [latex]y[\/latex]-intercept of [latex](0, 2)[\/latex] in the original function (blue line) reflects to the [latex]x[\/latex]-intercept [latex](2, 0)[\/latex] in the inverse function (green line). Also, the\u00a0[latex]x[\/latex]-intercept of [latex](\u20135, 0)[\/latex] in the original function (blue line) reflects to the [latex]y[\/latex]-intercept [latex](0, \u20135)[\/latex] in the inverse function (green line).<\/p>\n<p>The slopes of each function are also related. The function [latex]f(x)[\/latex] has a slope of [latex]\\frac{2}{5}[\/latex], while the inverse function [latex]f^{-1}(x)[\/latex] has a slope of [latex]\\frac{5}{2}[\/latex]. <strong>The slopes of inverse functions are reciprocals of each other. <\/strong>This is\u00a0because the slope of a function is [latex]\\dfrac{\\text{change in y}}{\\text{change in x}}[\/latex]. The slope of the inverse function becomes\u00a0[latex]\\dfrac{\\text{change in x}}{\\text{change in y}}[\/latex].<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-46\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>The Inverse of a Linear Function . <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Ex: Find an Inverse Function From a Table. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/TSztRfzmk0M\">https:\/\/youtu.be\/TSztRfzmk0M<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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